Mean temperature of the mixture = (T_{1} + T_{2}) / 2

Thus change in entropy is given by:

∆S = S_{2} – S_{1})

= mc ^{ (T1 + T2)/2}∫_{ T1} (dT / T) – mc ^{T2}∫_{ T1 + T2)/2} (dT / T)

= mc ln (T_{1} + T_{2})/2 T_{1}) –

mc ln (2 T_{2}) / (T_{1} + T_{2})

= mc ln (T_{1} + T_{2})/2 T_{1}) +

mc ln (T_{1} + T_{2}) /2 T_{2})

= mc ln (T_{1} + T_{2}) ^{2} / 4 T_{1} T_{2}

= mc ln [(T_{1} + T_{2}) / 2 (T_{1} T_{2}) ^{1/2}] ^{2}

= 2 mc ln [(T_{1} + T_{2}) / 2 (T_{1} T_{2}) ^{1/2}]

= 2 mc ln [(T_{1} + T_{2}) / 2 ]/[ (T_{1} T_{2})^{1/2}]

Hence, Resultant change of entropy of universe is:

2 mc ln [(T_{1} + T_{2}) / 2]/[ (T_{1} T_{2})^{1/2}]

The arithmetic mean (T_{1} + T_{2}) / 2 is greater than the geometric mean (T_{1} T_{2}) ^{1/2}

Therefore, ln [(T_{1} + T_{2}) / 2]/[ (T_{1} T_{2})^{1/2} ]

is always positive. Hence the entropy of the universe increases.